The category \({\Delta {T}\mathrm {op}}\)  carries a cofibrantly generated model structure, having the same generating cofibrations and generating trivial cofibrations as the standard model structure on topological spaces [1]}.
The geometric realisation functor \(|{-}| \colon {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\rightarrow {\Delta {T}\mathrm {op}}\)  takes values in \(\Delta \) -generated spaces, since \({\Delta {T}\mathrm {op}}\)  is closed under colimits of topological spaces and contains \(|\Delta ^n|\)  for each \(n \in \mathbb {N}_0\) .
By construction of the model structure on \({\Delta {T}\mathrm {op}}\) , the induced adjunction \(|{-}| : {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\rightleftarrows {\Delta {T}\mathrm {op}}: \operatorname{Sing}\)  is a Quillen adjunction (\(|{-}|\)  sends generating (trivial) cofibrations to (trivial) cofibrations).
Further, by the same proof as in [2]}, it follows that \(|{-}|\)  preserves finite products.
Then, the proof of [2]} applies as well, showing that \({\Delta {T}\mathrm {op}}\)  is a symmetric monoidal model category.
It also follows that \({\Delta {T}\mathrm {op}}\)  is a simplicial model category and that \(|{-}|\)  is a monoidal left Quillen functor.
Finally, the Quillen adjunction \(|{-}| \dashv \operatorname{Sing}\)  is even a Quillen equivalence, since the inclusion of \({\Delta {T}\mathrm {op}}\)  into Kelley spaces (or all topological spaces) is a Quillen equivalence [1]}; the claim then follows from the two-out-of-three property of Quillen equivalences.
